Typically, speakers like you have linked to in your last post will have a "key hole" extrusion on the back surface of the speaker. The speaker is placed onto a screw head with the larger part of the key hole and then slid down so the screw head holds the speaker on the smaller part of the extrusion.
If you can, try pushing up on the speaker as you pull them forward out of the cubby hole.
BTW, if it's only the speakers that have failed, it's not necessary to replace the whole unit. Replacement speakers can be installed and the headunit/amplifier/disc player retained.